A 31.0 kg child on a swing reaches a maximum height of 1.92 m above their rest position.

Assuming no loss of energy:
a) At what point during the swing will she attain their maximum speed?
b) What will be her maximum speed through the subsequent swing?
c) Assuming this maximum height was the result of one push from her parent, what was the

-at the bottom (max KE)

-max KE = max GPE
1/2 (31.0) v^2 = 584 J
v^2 = 37.67
v = 6.14 m/s
-idk what the last question is

so max KE = max GPE, I never thought of that. Can you explain this part?

bc at the bottom of her swing (KE=GPE), which equals 584 J

Since mass is irrelevant for part b I'm assuming they want the force of the push in part c which will be 584/1.92 (Fd =mgh)

To answer these questions, we can use the principles of conservation of mechanical energy. The mechanical energy of the child-swing system will be conserved as long as there is no loss of energy due to friction or other factors.

a) At what point during the swing will she attain their maximum speed?

The maximum speed will be attained at the lowest point, where the potential energy is at its minimum and the entire energy is in the form of kinetic energy. To find this point, we need to calculate the potential energy at the maximum height and equate it to the kinetic energy at the lowest point.

Potential energy at the maximum height (PE) is given by the equation:
PE = m * g * h
where m is the mass of the child (31.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (1.92 m).

Kinetic energy at the lowest point (KE) is given by the equation:
KE = (1/2) * m * v^2
where v is the velocity at the lowest point.

According to the conservation of mechanical energy, PE at the maximum height is equal to KE at the lowest point:
PE = KE

Using the equations mentioned above, we can solve for v:

m * g * h = (1/2) * m * v^2
v^2 = 2 * g * h
v = √(2 * g * h)

Substituting the values, we have:
v = √(2 * 9.8 * 1.92)
v ≈ 6.39 m/s

Therefore, the maximum speed will be attained at the lowest point of the swing (when the child is at a height of 0 meters) and its value will be approximately 6.39 m/s.

b) What will be her maximum speed through the subsequent swing?

As long as there's no energy loss, the maximum speed will remain the same throughout the swing. So the maximum speed during the subsequent swing will also be approximately 6.39 m/s.

c) Assuming this maximum height was the result of one push from her parent, what was the initial speed given to the child?

To find the initial speed given to the child by her parent, we need to consider the conservation of mechanical energy.

When the child reaches her maximum height, all the initial kinetic energy is converted into potential energy. So, the initial kinetic energy (KE_initial) is equal to the potential energy at the maximum height (PE).

Using the equation for potential energy:
PE = m * g * h

Substituting the values given:
PE = 31.0 kg * 9.8 m/s^2 * 1.92 m
PE ≈ 579.936 J

Since the initial kinetic energy is equal to the potential energy, the initial speed (v_initial) can be found using the equation for kinetic energy:

KE_initial = (1/2) * m * v_initial^2

v_initial^2 = (2 * KE_initial) / m
v_initial^2 = 2 * (579.936 J) / 31.0 kg
v_initial^2 ≈ 37.30 m^2/s^2

Taking the square root of both sides, we find:
v_initial ≈ √(37.30 m^2/s^2)
v_initial ≈ 6.11 m/s

Therefore, the initial speed given to the child by her parent is approximately 6.11 m/s.